contemporary methods of mortality analysis biodemography of mortality and longevity

CONTEMPORARY METHODS OF MORTALITY ANALYSIS

Biodemography of Mortality and Longevity

Leonid Gavrilov

Center on Aging NORC and the University of Chicago

Chicago, Illinois, USA

Demographics - 2013

Empirical Laws of Mortality

The Gompertz-Makeham Law

μ(x) = A + R e αx

A – Makeham term or background mortalityR e αx – age-dependent mortality; x - age

Death rate is a sum of age-independent component (Makeham term) and age-dependent component (Gompertz function), which increases exponentially with age.

risk of death

Gompertz Law of Mortality in Fruit Flies

Based on the life table for 2400 females of Drosophila melanogaster published by Hall (1969).

Source: Gavrilov, Gavrilova, “The Biology of Life Span” 1991

Gompertz-Makeham Law of Mortality in Flour Beetles

Based on the life table for 400 female flour beetles (Tribolium confusum Duval). published by Pearl and Miner (1941).

Source: Gavrilov, Gavrilova, “The Biology of Life Span” 1991

Gompertz-Makeham Law of Mortality in Italian Women

Based on the official Italian period life table for 1964-1967.

Source: Gavrilov, Gavrilova, “The Biology of Life Span” 1991

How can the Gompertz-Makeham law be used?

By studying the historical dynamics of the mortality components in this law:

μ(x) = A + R e αx

Makeham component Gompertz component

Historical Stability of the Gompertz

Mortality ComponentHistorical Changes in Mortality for 40-year-old

Swedish Males1. Total mortality, μ40

2. Background mortality (A)

3. Age-dependent mortality (Reα40)

Source: Gavrilov, Gavrilova, “The Biology of Life Span” 1991

The Strehler-Mildvan Correlation:

Inverse correlation between the Gompertz

parametersLimitation: Does not take into

account the Makeham parameter that leads to spurious correlation

Modeling mortality at different levels of Makeham parameter but

constant Gompertz parameters

1 – A=0.01 year-1

2 – A=0.004 year-1

3 – A=0 year-1

Coincidence of the spurious inverse correlation between the Gompertz parameters

and the Strehler-Mildvan correlation

Dotted line – spurious inverse correlation between the Gompertz parameters

Data points for the Strehler-Mildvan correlation were obtained from the data published by Strehler-Mildvan (Science, 1960)

Compensation Law of Mortality(late-life mortality

convergence) Relative differences in death

rates are decreasing with age, because the lower initial death rates are compensated by higher slope (actuarial aging rate)

Compensation Law of Mortality

Convergence of Mortality Rates with Age

1 – India, 1941-1950, males 2 – Turkey, 1950-1951,

males3 – Kenya, 1969, males 4 - Northern Ireland, 1950-

1952, males5 - England and Wales,

1930-1932, females 6 - Austria, 1959-1961,

females 7 - Norway, 1956-1960,

females

Source: Gavrilov, Gavrilova,“The Biology of Life Span”

1991

Compensation Law of Mortality (Parental Longevity Effects)

Mortality Kinetics for Progeny Born to Long-Lived (80+) vs Short-Lived Parents

Sons DaughtersAge40 50 60 70 80 90 100

Log(

Haz

ard

Rat

e)

0.001

0.01

0.1

1

short-lived parentslong-lived parents

Linear Regression Line

Age40 50 60 70 80 90 100

Log(

Haz

ard

Rat

e)

0.001

0.01

0.1

1

short-lived parentslong-lived parents

Linear Regression Line

Compensation Law of Mortality in Laboratory

Drosophila1 – drosophila of the Old

Falmouth, New Falmouth, Sepia and Eagle Point strains (1,000 virgin females)

2 – drosophila of the Canton-S strain (1,200 males)

3 – drosophila of the Canton-S strain (1,200 females)

4 - drosophila of the Canton-S strain (2,400 virgin females)

Mortality force was calculated for 6-day age intervals.

Source: Gavrilov, Gavrilova,“The Biology of Life Span”

1991

ImplicationsBe prepared to a paradox that higher

actuarial aging rates may be associated with higher life expectancy in compared populations (e.g., males vs females)

Be prepared to violation of the proportionality assumption used in hazard models (Cox proportional hazard models)

Relative effects of risk factors are age-dependent and tend to decrease with age

The Late-Life Mortality Deceleration (Mortality Leveling-off,

Mortality Plateaus)

The late-life mortality deceleration law states that death rates stop to increase exponentially at advanced ages and level-off to the late-life mortality plateau.

Mortality deceleration at advanced ages.

After age 95, the observed risk of death [red line] deviates from the value predicted by an early model, the Gompertz law [black line].

Mortality of Swedish women for the period of 1990-2000 from the Kannisto-Thatcher Database on Old Age Mortality

Source: Gavrilov, Gavrilova, “Why we fall apart. Engineering’s reliability theory explains human aging”. IEEE Spectrum. 2004.

Mortality Leveling-Off in House Fly

Musca domestica

Based on life table of 4,650 male house flies published by Rockstein & Lieberman, 1959

Age, days0 10 20 30 40

haza

rd ra

te, l

og s

cale

0.001

0.01

0.1

Mortality Deceleration in Animal Species

Invertebrates: Nematodes, shrimps,

bdelloid rotifers, degenerate medusae (Economos, 1979)

Drosophila melanogaster (Economos, 1979; Curtsinger et al., 1992)

Housefly, blowfly (Gavrilov, 1980)

Medfly (Carey et al., 1992) Bruchid beetle (Tatar et al.,

1993) Fruit flies, parasitoid wasp

(Vaupel et al., 1998)

Mammals: Mice (Lindop, 1961;

Sacher, 1966; Economos, 1979)

Rats (Sacher, 1966) Horse, Sheep, Guinea

pig (Economos, 1979; 1980)

However no mortality deceleration is reported for

Rodents (Austad, 2001) Baboons (Bronikowski

et al., 2002)

Existing Explanations of Mortality Deceleration

Population Heterogeneity (Beard, 1959; Sacher, 1966). “… sub-populations with the higher injury levels die out more rapidly, resulting in progressive selection for vigour in the surviving populations” (Sacher, 1966)

Exhaustion of organism’s redundancy (reserves) at extremely old ages so that every random hit results in death (Gavrilov, Gavrilova, 1991; 2001)

Lower risks of death for older people due to less risky behavior (Greenwood, Irwin, 1939)

Evolutionary explanations (Mueller, Rose, 1996; Charlesworth, 2001)

Testing the “Limit-to-Lifespan” Hypothesis

Source: Gavrilov L.A., Gavrilova N.S. 1991. The Biology of Life Span

Implications There is no fixed upper limit to human longevity - there is no special fixed number, which separates possible and impossible values of lifespan.

This conclusion is important, because it challenges the common belief in existence of a fixed maximal human life span.

Latest Developments

Was the mortality deceleration law overblown?

A Study of the Extinct Birth Cohorts in the United States

More recent birth cohort mortality

Nelson-Aalen monthly estimates of hazard rates using Stata 11

What about other mammals?

Mortality data for mice: Data from the NIH Interventions Testing

Program, courtesy of Richard Miller (U of Michigan)

Argonne National Laboratory data, courtesy of Bruce Carnes (U of Oklahoma)

Mortality of mice (log scale) Miller data

Actuarial estimate of hazard rate with 10-day age intervals

males females

Alternative way to study mortality trajectories at advanced ages:

Age-specific rate of mortality change

Suggested by Horiuchi and Coale (1990), Coale and Kisker (1990), Horiuchi and Wilmoth (1998) and later called ‘life table aging rate (LAR)’

k(x) = d ln µ(x)/dx

Constant k(x) suggests that mortality follows the Gompertz model. Earlier studies found that k(x) declines in the age interval 80-100 years suggesting mortality deceleration.

Age-specific rate of mortality change

Swedish males, 1896 birth cohort

Flat k(x) suggests that mortality follows the Gompertz law

Age, years

60 65 70 75 80 85 90 95 100

k x v

alue

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

Study of age-specific rate of mortality change using cohort

data

Age-specific cohort death rates taken from the Human Mortality Database

Studied countries: Canada, France, Sweden, United States

Studied birth cohorts: 1894, 1896, 1898k(x) calculated in the age interval 80-100 yearsk(x) calculated using one-year mortality rates

Slope coefficients (with p-values) for linear regression models of k(x)

on age

All regressions were run in the age interval 80-100 years.

Country Sex Birth cohort

1894 1896 1898

slope p-value slope p-value slope p-value

Canada F -0.00023 0.914 0.00004 0.984 0.00066 0.583

M 0.00112 0.778 0.00235 0.499 0.00109 0.678

France F -0.00070 0.681 -0.00179 0.169 -0.00165 0.181

M 0.00035 0.907 -0.00048 0.808 0.00207 0.369

Sweden F 0.00060 0.879 -0.00357 0.240 -0.00044 0.857

M 0.00191 0.742 -0.00253 0.635 0.00165 0.792

USA F 0.00016 0.884 0.00009 0.918 0.000006 0.994

M 0.00006 0.965 0.00007 0.946 0.00048 0.610

What are the explanations of mortality

laws?

Mortality and aging theories

What Should the Aging Theory Explain Why do most biological species

including humans deteriorate with age?

The Gompertz law of mortality

Mortality deceleration and leveling-off at advanced ages

Compensation law of mortality

Additional Empirical Observation:

Many age changes can be explained by cumulative effects of

cell loss over time Atherosclerotic inflammation -

exhaustion of progenitor cells responsible for arterial repair (Goldschmidt-Clermont, 2003; Libby, 2003; Rauscher et al., 2003).

Decline in cardiac function - failure of cardiac stem cells to replace dying myocytes (Capogrossi, 2004).

Incontinence - loss of striated muscle cells in rhabdosphincter (Strasser et al., 2000).

Like humans, nematode C. elegans experience muscle loss

Body wall muscle sarcomeres

Left - age 4 days. Right - age 18 days

Herndon et al. 2002. Stochastic and genetic factors influence tissue-specific decline in ageing C. elegans. Nature 419, 808 - 814.

“…many additional cell types (such as hypodermis and intestine) … exhibit age-related deterioration.”

What Is Reliability Theory? Reliability theory is a general

theory of systems failure developed by mathematicians:

Aging is a Very General Phenomenon!

Stages of Life in Machines and Humans

The so-called bathtub curve for technical systems

Bathtub curve for human mortality as seen in the U.S. population in 1999 has the same shape as the curve for failure rates of many machines.

Gavrilov, L., Gavrilova, N. Reliability theory of aging and longevity. In: Handbook of the Biology of Aging. Academic Press, 6th edition, 2006, pp.3-42.

The Concept of System’s Failure

In reliability theory failure is defined as the event when a required function is terminated.

Definition of aging and non-aging systems in reliability

theoryAging: increasing risk of failure

with the passage of time (age). No aging: 'old is as good as new'

(risk of failure is not increasing with age)

Increase in the calendar age of a system is irrelevant.

Aging and non-aging systems

Perfect clocks having an ideal marker of their increasing age (time readings) are not aging

Progressively failing clocks are aging (although their 'biomarkers' of age at the clock face may stop at 'forever young' date)

Mortality in Aging and Non-aging Systems

Age0 2 4 6 8 10 12

Ris

k of

dea

th

1

2

3

Age0 2 4 6 8 10 12

Ris

k of

Dea

th

0

1

2

3

non-aging system aging system

Example: radioactive decay

According to Reliability Theory:Aging is NOT just growing oldInsteadAging is a degradation to failure: becoming sick, frail and dead

'Healthy aging' is an oxymoron like a healthy dying or a healthy disease

More accurate terms instead of 'healthy aging' would be a delayed aging, postponed aging, slow aging, or negligible aging (senescence)

The Concept of Reliability Structure

The arrangement of components that are important for system reliability is called reliability structure and is graphically represented by a schema of logical connectivity

Two major types of system’s logical connectivity

Components connected in series

Components connected in parallel

Fails when the first component fails

Fails when all

components fail

Combination of two types – Series-parallel system

Ps = p1 p2 p3 … pn = pn

Qs = q1 q2 q3 … qn = qn

Series-parallel Structure of Human Body

• Vital organs are connected in series

• Cells in vital organs are connected in parallel

Redundancy Creates Both Damage Tolerance and Damage Accumulation

(Aging)

System with redundancy accumulates damage (aging)

System without redundancy dies after the first random damage (no aging)

Reliability Model of a Simple Parallel

SystemFailure rate of the

system:

Elements fail randomly and independently with a constant failure rate, k

n – initial number of elements

nknxn-1 early-life period approximation, when 1-e-kx kx k late-life period approximation, when 1-e-kx 1

( )x = dS( )xS( )x dx

= nk e kx( )1 e kx n 1

1 ( )1 e kx n

Failure Rate as a Function of Age in Systems with Different Redundancy

Levels

Failure of elements is random

Standard Reliability Models Explain

Mortality deceleration and leveling-off at advanced ages

Compensation law of mortality

Standard Reliability Models Do Not Explain

The Gompertz law of mortality observed in biological systems

Instead they produce Weibull (power) law of mortality growth with age

An Insight Came To Us While Working With Dilapidated

Mainframe Computer The complex

unpredictable behavior of this computer could only be described by resorting to such 'human' concepts as character, personality, and change of mood.

Reliability structure of (a) technical devices and (b) biological

systems Low redundancy

Low damage load

High redundancy

High damage load

X - defect

Models of systems with distributed redundancy

Organism can be presented as a system constructed of m series-connected blocks with binomially distributed elements within block (Gavrilov, Gavrilova, 1991, 2001)

Model of organism with initial damage load

Failure rate of a system with binomially distributed redundancy (approximation for initial period of life):

x0 = 0 - ideal system, Weibull law of mortality x0 >> 0 - highly damaged system, Gompertz law of

mortality

( )x Cmn( )qk n 1 qqk

x + n 1

= ( )x0 x + n 1

where - the initial virtual age of the systemx0 =1 qqk

The initial virtual age of a system defines the law of system’s mortality:

Binomial law of mortality

People age more like machines built with lots of faulty parts than like ones built with

pristine parts. As the number

of bad components, the initial damage load, increases [bottom to top], machine failure rates begin to mimic human death rates.

Statement of the HIDL hypothesis:

(Idea of High Initial Damage Load )

"Adult organisms already have an exceptionally high load of initial damage, which is comparable with the amount of subsequent aging-related deterioration, accumulated during the rest of the entire adult life."

Source: Gavrilov, L.A. & Gavrilova, N.S. 1991. The Biology of Life Span: A Quantitative Approach. Harwood Academic Publisher, New York.

Why should we expect high initial damage load in biological systems?

General argument:--  biological systems are formed by self-assembly without helpful external quality control.

Specific arguments:

• Most cell divisions responsible for  DNA copy-errors occur in early development leading to clonal expansion of mutations• Loss of telomeres is also particularly high in early-life• Cell cycle checkpoints are disabled in early development

Practical implications from the HIDL hypothesis:

"Even a small progress in optimizing the early-developmental processes can potentially result in a remarkable prevention of many diseases in later life, postponement of aging-related morbidity and mortality, and significant extension of healthy lifespan."

Source: Gavrilov, L.A. & Gavrilova, N.S. 1991. The Biology of Life Span: A Quantitative Approach. Harwood Academic Publisher, New York.

Month of BirthJan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

life

expe

ctan

cy a

t age

80,

yea

rs

7.6

7.7

7.8

7.9

1885 Birth Cohort1891 Birth Cohort

Life Expectancy and Month of Birth

Data source: Social Security Death Master File

Conclusions (I) Redundancy is a key notion for

understanding aging and the systemic nature of aging in particular. Systems, which are redundant in numbers of irreplaceable elements, do deteriorate (i.e., age) over time, even if they are built of non-aging elements.

An apparent aging rate or expression of aging (measured as age differences in failure rates, including death rates) is higher for systems with higher redundancy levels.

Conclusions (II) Redundancy exhaustion over the life course explains the

observed ‘compensation law of mortality’ (mortality convergence at later life) as well as the observed late-life mortality deceleration, leveling-off, and mortality plateaus.

Living organisms seem to be formed with a high load of initial damage, and therefore their lifespans and aging patterns may be sensitive to early-life conditions that determine this initial damage load during early development. The idea of early-life programming of aging and longevity may have important practical implications for developing early-life interventions promoting health and longevity.

AcknowledgmentsThis study was made possible

thanks to: generous support from the

National Institute on Aging (R01 AG028620) Stimulating working environment at the Center on Aging, NORC/University of Chicago

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